Le Rhesus (Simia Rhesus) – illustration by Jean-Baptiste Audebert for his treatise, Histoire naturelle des singes et des makis (1799) – image: Bibliothèque nationale de France / gallica.bnf.fr

This word is from French rhésus, formerly rhesus, and from its etymon, the scientific Latin (Simia) Rhesus.

In Histoire naturelle des singes et des makis (Natural History of the Monkeys and Lemurs – 1799), the French artist and naturalist Jean-Baptiste Audebert (1759-1800) gave to a small brown macaque with red skin on the face and rump, native to southern Asia, the scientific Latin name Simia Rhesus and the French name rhesus:

Le nom de Macaque à queue courte, imposé par Buffon à cet animal, n’est pas un nom, mais une phrase, qui semble indiquer une simple variété du Macaque. Il me semble qu’au contraire ce singe doit être considéré comme une espèce très-distincte […] En conséquence, je le nommerai Simia Rhesus ; ce nom, n’ayant aucune signification, peut être appliqué à ce singe, et éloigner toute idée de rapports spécifiques entre lui et les autres espèces.
Le caractère par lequel on distingue le Rhesus des autres espèces est sa queue courte [etc.]
translation:
The name of short-tailed Macaque, imposed by Buffon on this animal, is not a name, but a phrase, which seems to indicate a mere variety of the Macaque. It seems to me that on the contrary this monkey must be considered a very distinct species […] As a consequence, I will name it Simia Rhesus; this name, having no signification, may be applied to this monkey, and dismiss any idea of specific relationships between it and the other species.
The characteristic by which one distinguishes the Rhesus from the other species is its short tail [etc.]

The adoption by Audebert of this name for the monkey was therefore arbitrary. In classical Latin, Rhesus, from ancient Greek Ῥῆσος (= Rhesos), was the name of a mythical king of Thrace, who arrived in the tenth year of the Trojan War to aid Troy. Odysseus and Diomedes stole his horses because an oracle had said that if these horses drank from the River Xanthus, Troy would not fall.

The scientific name of the rhesus macaque, or rhesus monkey, is now Macaca mulatta, family Cercopithecidae. It is often kept in captivity and is widely used in medical research.

Since the 1940s, rhesus, abbreviated as Rh, has been applied to a major human blood group system originally identified by means of rabbit antisera to rhesus monkey blood, consisting of three principal antigens (one major, D, which is the most immunogenic, and two minor, C and E).